Step 1: Understanding Animation Timeline Architecture:
In 3D animation suites such as Autodesk Maya, the time-control interface (Time Slider and Range Slider) is split into two distinct time range limits. This division allows animators to manage projects with high levels of complexity by isolating specific frame segments during design.
Step 2: Defining Frame Range (Animation Global Range):
The Frame Range (commonly termed the global or animation range) represents the absolute start and end frames of the active scene timeline. It defines the entire span of frames available for setting keyframes and rendering the complete sequence. For example, a 1-minute shot running at $24\,\text{fps}$ has an absolute global Frame Range from frame $1$ to frame $1440$.
Step 3: Defining Playback Range (Active Local Range):
The Playback Range (or active playback range) is a localized segment within the larger global Frame Range. It represents the specific frames that play and loop in the active viewport when you press the play button. If an animator is working on a short, 3-second jump cycle between frames $120$ and $192$ within a $1440$-frame sequence, they set the Playback Range limits to $120$ and $192$. This allows them to loop and evaluate just that motion sequence without waiting for the entire scene to play.
Step 4: Mathematical Representation of Timeline Ranges:
Let $F_{\text{global\_start}}$ and $F_{\text{global\_end}}$ define the limits of the Frame Range. Let $F_{\text{play\_start}}$ and $F_{\text{play\_end}}$ define the Playback Range. These parameters must strictly satisfy the following inequality:
$$F_{\text{global\_start}} \le F_{\text{play\_start}} < F_{\text{play\_end}} \le F_{\text{global\_end}}$$
This inequality mathematically guarantees that the active playback range always stays within the absolute boundaries of the global animation sequence.